Generalized eigenvector
is a vector which satisfies certain criteria which are more relaxed than those for an (ordinary) eigenvector.
generate a Jordan chain of linearly independent generalized eigenvectors which form a basis for an invariant subspace of
[24] Diagonalizable matrices are of particular interest since matrix functions of them can be computed easily.
linearly independent generalized eigenvectors associated with it and can be shown to be similar to an "almost diagonal" matrix
[33] These results, in turn, provide a straightforward method for computing certain matrix functions of
has real-valued elements, then it may be necessary for the eigenvalues and the components of the eigenvectors to have complex values.
Notice that this matrix is in Jordan normal form but is not diagonal.
Since there is one superdiagonal entry, there will be one generalized eigenvector of rank greater than 1 (or one could note that the vector space
are linearly independent and hence constitute a basis for the vector space
Unfortunately, it is a little difficult to construct an interesting example of low order.
Together the two chains of generalized eigenvectors span the space of all 5-dimensional column vectors.
[44] Definition: A set of n linearly independent generalized eigenvectors is a canonical basis if it is composed entirely of Jordan chains.
Thus, once we have determined that a generalized eigenvector of rank m is in a canonical basis, it follows that the m − 1 vectors
designates the number of linearly independent generalized eigenvectors of rank k corresponding to the eigenvalue
linearly independent generalized eigenvectors of a canonical basis for the vector space
We now define Consequently, there will be three linearly independent generalized eigenvectors; one each of ranks 3, 2 and 1.
Note that it is possible to obtain infinitely many other generalized eigenvectors of rank 3 by choosing different values of
as generalized eigenvectors of rank 2 and 1, respectively, where and The simple eigenvalue
can be dealt with using standard techniques and has an ordinary eigenvector A canonical basis for
is an n × n matrix whose columns, considered as vectors, form a canonical basis for
[50] Note that some textbooks have the ones on the subdiagonal, that is, immediately below the main diagonal instead of on the superdiagonal.
in Jordan normal form, obtained through the similarity transformation
Find a matrix in Jordan normal form that is similar to Solution: The characteristic equation of
will contain one linearly independent generalized eigenvector of rank 2 and two linearly independent generalized eigenvectors of rank 1, or equivalently, one chain of two vectors
[55] In Example 3, we found a canonical basis of linearly independent generalized eigenvectors for a matrix
[60] Using generalized eigenvectors, we can obtain the Jordan normal form for
and these results can be generalized to a straightforward method for computing functions of nondiagonalizable matrices.
Consider the problem of solving the system of linear ordinary differential equations where If the matrix
Continuing this procedure, we work through (9) from the last equation to the first, solving the entire system for
these functions solve the system of equations, Proof: Define Then, as
